Equipartition of energy and quantization

A mechanical system consisting of N particles is described by specifying three coordinates for each particle, or a total of 3N coordinates. Thus there are 3N independent components of the motion, or degrees of freedom, in such a system. If the N particles are bound together to form a polyatomic molecule, then the 3N coordinates and components of the motion are conveniently chosen as follows. 

Translational. Three coordinates describe the position of the center of mass motion in these coordinates corresponds to translation of the molecule as a whole. The energy stored in this mode of motion is kinetic energy only, trans = + jmv + 

Each of these terms contains the square of a velocity component and therefore, as we have seen previously, each contributes jkT to the average energy. 

Rotational. Two angles are needed to describe the orientation of a linear molecule in space three angles are needed for the description of the orientation of a nonlinear molecule. Motion in these coordinates corresponds to rotation about two axes (linear molecule) or three axes (nonlinear molecule) in space. The equation for the energy of rotation has the forms 

Vibrational. There remain 3N — 5 coordinates for linear molecules and 3N — 6 coordinates for nonlinear molecules. These coordinates describe the bond distances and bond angles within the molecule. Motion in these coordinates corresponds to the vibrations (stretching or bending) of the molecule. Thus linear molecules have 3N — 5 vibrational modes nonlinear molecules have 3N — 6 vibrational modes. Assuming that the vibrations are harmonic, the energy of each vibrational mode can be written in the form 

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